96 Theory of Relativity
close to the speed of light the description is provided by the theory of
relativity.
Special relativity mechanics arose from Newtonian mechanics and the
hypothesis that the mathematical equations describing laws of mechanics
should have the same form in every two inertial frames that move with the
constant velocity relative to each other. This hypothesis of mathematical
invariance does not just express a mathematical aesthetic. From a physical
standpoint, the hypothesis requires that two observers, one in a frame O
and the other in a frame 0\ moving with constant velocity v relative to
O, should be able to describe the motion of a particle using the same
law. One such law could be Newton's Second Law (2.1.1) on page 40.
Although Newton understood the position is relative to the observer in
different frames, he assumed time to be an absolute physical quantity that
is the same in different frames. Mathematically, these assumptions are
expressed by the Galilean transformation ri = r + vt, t\ = t. Note
that the Galilean transformation does leave (2.1.1) unchanged: f\ = r,
and so if m r = F, then m'f\ = F.
The assumption of absolute time contradicts the experimental result,
not known at Newton's time, that light has the same speed in O and 0.
We will see that the constant speed of light implies that time is as rela-
tive as position, and the Galilean transformation must be replaced by that
of Lorentz. Not surprisingly, the Lorentz transformation no longer leaves
(2.1.1) unchanged, and we will see how the law of motion must be modified
to satisfy the invariance hypothesis; see (2.4.15) on page 102 below. This
new law of motion is a more general mathematical model of mechanics than
(2.4.1) and leads to a number of remarkable conclusions, such as the famous
relation £ — mc^2 between the mass and energy.
The theory of relativity is universally considered as one of the pinnacles
of human intellectual achievements. Many also believe that understanding
of this theory is beyond the ability of all but very few bright minds. Our
main objective in the following few sections is to demystify the subject and
to demonstrate that one only needs a basic knowledge of linear algebra
and multi-variable calculus to understand the main ideas of the theory, to
follow the computations that lead to the most famous implications of this
theory, and to see how the mathematical models of relativity interact with
the underlying physical phenomena.
We start with a short historical background, mostly related to the de-
velopment of special relativity. Next, we present an elementary derivation
of the Lorentz transformation, and finally discuss the main results of gen-